Logarithmic scale.



ii SHEETS-SHEET 1.

Patented Aug. 22, 1911.

J. MGC. MICHAELSON. LOGARITHMIG SCALE.

APPLIUATION FILED 11111.12, 1907.

kan@ J. M00. MIGHAELSON.v

100111111111110 SCALE. APPLIOATION 'ILED 11113.12, 1907. y

Patented Aug.'22, 1911.

5 SHEETS-SHEET 2.

human@ l x fsqvm J. 'M00'. MICHAELsoN.

L00111111.11111110 SCALE. APPLIOATION FILED MAB. 12, 1907.

Patented' Aug. 22, 1911.

5 SHEETS-SHEET 3.

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. J. M00. MIGHAELSON.

LOGARITHMIC SCALE.

APPLICATION FILED MAR. 12, 1907.

1,001,061. Patented Aug. 22, 1911..l

J. MCC. MIGHAELSON.

LOGARITHMIG SCALE.

APPLICATION FILED 11.111.12.190?.

1,001,061, Patented Aug.22,1911.

5 SHEETS-SHEET 6.

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JOSEPH MCC. MICHAELSON, 0F GENEVA, NEW YORK.

LOGARITHMIC SCALE.

Specification of Letters Patent.

Patented Aug. 22, 1911.

Application led March 12, 1907. Serial No. 361,915.

ATo all 'whom it may concern:

Be it kiown that I, JOSEPH MUC. MI- oHAEiJsoN, a citizen of the United States, residing at Geneva, in the county .of .Ontario and State of New York, have invented certain new and useful Improvements in Loga- ,rithmic Scales, of which the following is a specification, reference being had therein to the accompanying drawing.

'This invention relates to certain new and useful improvements in the construction of calculating or slide rules, particularly adapted for use in solving problems involved in designing and in investigating the design of reinforced concrete.

The object of the present invention is the provision of a scale or rule of this, character for mechanically calculating problems involvedin designing 'and in investigating the fdesign of reinforced concrete beams, by

` which-with certaingiven factors the other 4factors may be readily determined without lthe necessity of calculation or reference to rules, tables, or data, other than thosefurnislied by the rule in use.

-For the purposes of the specification the term beams is used in the sense of structures subjected to transverse loads or eccen-A tric forces. I The invention comprises broadly a rule provided with a series of scales indicating vthe factors laid out in distances-proportional to the logarithins of each series of factors, Icertain of said scales being movable with reference to certain others of the scales,

so that their adjustment will effect an addivstruction of slide rules.

'tion or subtraction of the logarithms of the culations linvolved by the formulae, and

have, therefore, the drawings, accompanying this specification, shown several different forms'of scalesfor carrying out the invention.

refer to similar parts, Figure 1 is a plan view of one side of my improved rule, showing the saine in position for use in the solution of certain problems as hereinafter described and demonstrated. Fig. 2 is a plan; View of Referring now more l.particularly tov the v drawings', wherein like numerals of reference.

the reverse side of the rule illustrated in Fig. 1. Fig. 3 is a plan view of a modiied form of the invention in which all of lthe scales are confined to one side of the rule. Fig. 4 is a cross section on line 4.-4 of Fig. 1. Fig. 5 1s a cross section on line A5 5 of Fig. 3. Fig. 6 is a modiied form of the invention as applied to a rotary rule. Fig. 7 is a cross section of the same, and Fig. 8 is a modified form of Fig. 1, showing a variation in the scale arrangement.

While the rule may be constructed in any suitable .manner, I prefer to construct the same as shown in Figs. 12 and 3, in which, 1 '1 designates separated square edged side pieces, preferably formed of laminated material to prevent warping thereof. These pieces are rigidly connected at their ends by metallic cross pieces 2 2 secured to the upper and lower faces thereof. The upper and lower faces of the side pieces 1 1 are covvered preferably by celluloid sheets 3 3,

which are secured in position by any convenient means, and upon' which are desifrned to be marked certain scales hereinafler referred to. The sheets 3 3 on the rear are provided with suitable slots or openings therein, so that the sliding portion of the scale which works between the side pieces 1 1 may be observed and brought into proper position when the rule isin use. The slide comprises a member 4, which fits between the side pieces 1 1, and the edges of which are squared and fit snugly against the squared edges of said side pieces. Displacement of the slide 4 bytilting is prevented by ymeans of the overhanging celluloidplates inafter set forth. The square` edges of the y may be observed. 4The edges of said open? crete. Marke on the plate ofcelluloid or other material secured to the upper face of the slide 4, so as to coperate wit the scales A and F hereinbefore referred to, is a scale C, indicating span in feet and depth of concrete in inches, and also lmarked on the up` per face of the slide 4. so as to coperate with the scales B and E hereinbefore referred to, is a scale D indicatin area of steel in square inches per foot wi e of concrete. The u per face of the slide 4 has also marked At ereon a scale'C, indicating pounds per uare inch on concrete. The scales A, B, C, D, Eand E are logarithmic scales-'that is, scales in which the distance of any number from the unit of the scale corres onds to the mantissa of the logarithm of t at number and such scales are divided intoas many part-s. as is desired, the number and degree of neness of the subdivisions being optional with the manuf facturer. The lluloid or other plate 3 secured over the bottom faces of the side pieces 1-1 is also provided mediall thereof Withan opening through which t e lower face of the slide 4 ma be observed. The portion of the celluloi sheet formin one edge of this opening is straight, an has marked thereon ascale indicatxn standard commercial sizes of round an square rods used inconcrete reinforced-work an'd also a scale G indicating special types of bars sometimes used when discontinuous or separated material is employedin reinforcement of concrete. A portion of the parts 1v-1 and celluloid sheet 3 forming the ack piece'of the body of the rule iis formed in several straight open sectionsgwhich are' offset from each other one"of which is provided at its edge with scale G heretofore referred to,`ind1cating commercial sizes of rods used in reinforce work. Another of said straight portions is provided at its edge4 with a scale Iv thereon, indicatin different center that any or -all of the rods listed therein must be placed to equal the area of V steel shown to be required on scale of areas D', hereinbefore referred to. Y The lower face tov hereinbefore referred to, both scales indicating the distance from center to of the slide 4 is also provided with a scale of ratios J co erating with the scale I, hereinbefore re erred to, to determine the size of expanded metal necessary to be used ive the desired ercenta e as shown on sca e of ratios A or herein fore referred to. It will be obvious that by forming one edge of theV celluloid or other sheet in a stepped form as shown in Fig. 2 that several different scales maybe readily marked -thereon5 and also that the'olset portion so formed serves to conceal the figures of the scale H, which are not in use, thereby lessening the liability of the user to make an incorrect readin The combination of scales .on this face o the rule is so arranged relatively to that on the upper face thereof (shown in Fig. 1), that the area on the scale' of areas D directly over the member 32 of the scale of ratios B will be found equal to any selected size or type shown on scales G, G and I, if such selection be spaced apart the distance from center tocenter indicated by the portion of the scales H H and J immediately opposite such selected' Should a ratio or percenta e otherjthanA 32 be used, the slide 4 shou d be moved 'to bring the area over the vselected ratio or percentage opposite the number 32 in the scale of ratios B, when the .reading as to the distance a art which the reinforce rods must be place to give the desired area may be made as before. By offsetting the open-- ings itis possible to use a greater number of scales than is possible with slide rules of the usual combination: Also by this means, error in reading results is largel avoided in that scales not involved in the immediate transaction are removed from si ht.

. In the form of my invention d1sclosed in Fig. 3 of the drawing theside is royided with the same scales as heretofore escribed with reference to Figs however arranged slightlyldiferentin order that all of the necessa 'readin'v -xnade be made from one side o the; u e. In this' form of. my 4invention the rule comprises a base 6, which is preferably-'formedof laminated material, and which is provided with a channel extending longitudlnally thereof' having uareded which abut the squared side wal s of sai channel, the slide being retained in position by a late 8 of celluloi or other suitable materia which is secured to the top of the base in any suitable manner and by metallic strips 9,.extending across the ends thereofl The plate 8 has formed Itherein an openin extending substantially the full len h o the rule and two openings O, O", z, extending for-a portion of the length of the rule, which are offset with reference to each other. The slide 7 and late 8 are provided with scales A, B, C, 2, E, F, G, G', H, H', which are identical in relative arrangementsv and functions with 1 and 2, which are I lao the scales similarly designated and in connection with Figs. 1 and 2 and need not be therefore again described in detail.

In the form of my invention disclosed in Fig. 6, the rule comprises a base 10 and a disk'11,'which is mounted to rotate on said base, the base being provided with a portion T projecting beyond the periphery of the disk 11, and the disk being provided with out away portions T, T2, permitting the inspection of certain scales on the base therebeneath. The disk and base are provided with scales A, B, C, D, E, F, G, G',

H, H', which are identical in relative ar- 'In the designing of, and in the investigation of desi sof reinforced concrete, the factors invo ved are, the load, span, unit stresses, moduli of elasticity, thickness of concrete and the area. 'of embedded metal. There are various formulae for determining these factors. Those in general use are known as the straight line and the parabolic formul, both of which are again modified by designers, some of whom use the ultimate and elastic limit values for the unit stresses; thus determining the ultimate strength. While others prefer the use of safe unit stresses. with the factor of safety in the unit stresses used, there being also other variations sometimes used. In the solution -of problems involving the use of any orall of these methods, my rule isv equally well adapted.

Referring to the accompanying drawings, Figs. 1, 2 and 3 illustrating my rule and lts uses, both the straight. line and the parabolic formulae are employed, Naviers theory and thel theory of elacticity being used in the development of both formulae. In the straight line formulae used all tension in the concrete Ais omitted. Tension in the concrete isl usually omitted in all formulae, but to illustrate the scope of the use of this rule, it is not so omitted in the parabolic formulae illustrated. The safe unit stresses are used in the straight formulae illustrated. The ultimate unit stress in the concrete and the unit stress at elastic limit in the steel are used in the parabolic formulae.- Moduli corresponding4 to theunit stresses p used are employed in the straight line formulas, while in tlie parabolic ormul the ultimate modulus is assumed to be 2,/3 of the initial modulus of the concrete. As will be seen by these'citations of extremes, my rule 'is equally yvell adapted to the solution of problems based on any formulae. The formulae in their most general form are as folpRd in which R E Modulus of elasticity of steel E."Initial modulus of elasticity of concrete s Stress per square unit in steel c "Stress per square unit in concrete d-depth from outer liber in compression to center of steel in tension.

:1w-distance from outer liber in com ression to neutral axis.

a-'ratio of modulus of elasticity o concrete at outer ber to initial modulus.

1J-ratio of modulus of elasticity of concrete at ultimate to initial modulus.

u-ratio of deformation of concrete at outer fiber to deformation at ultimate.

b-breadth-oi concrete.

A-area of steel in tension for breadth, b.

g-percentage of steel reinforcement for area bd of concrete.

distance of center of gravity of total force in compression from the outer fiber in compression.

c-Summatiom of all forces in compression.

X-factot if tension is considered in the concrete-1 if tension I ln concrete is not considered. lil-factor due to moment of tension of concrete, if considered about z=o if tension is not considered.

. Ki=factor due to moment o! tension about center of stee1=o if tension is not considered.

M.=moment of tensile forces about center of gravity of compressive forces.

ISL-moment of compressive forces, concrete in tension, about center of steel.

For the straight line formulae, with all tenslon taken by the steel, these equatlons become the following For the parabolic formulae with ultimate l value for C and tension in lthe concrete :1, a:a:any fraction between the limits 1/2 and 1. Usually taken 1/2 K: 0.98 K1:1/150, K2:1/75. These values substituted in the general equations give the required formulae, which are similar in form to the straight line. For the arabolic formulae with ultimate value of and all tension taken by the steel and the ultimate value of the modulus of4 elasticity for concrete used in determining the ratio, the

new 1value of this ratio is times that given in the general equations z'. e. R:aR, in which R is the new ratio and u:1, a:a:l/2 generally. In these equations the moment M of the internal forces being .equal to the moment of the external forces,

.ama/a I A=nd ,II Transforming above equations I and II to logarithmic forms we get the following: Log. 1 Z:log. m-i-log. Z-i-1/2 log. to Y Log. A:log. n-i-log. d On the scale of loads F, 1/2 log. fw is laid out. On the scale Cy of spans log'. Z

is laid out and the "same points used as a scale of thickness of concrete. m is computed and the log. is laid out on scale A. It will be observed that thisi`method of laying out one scale for two separate and distinct purposes having one series of figures for the sub-divisions,-in this case feet with decimal arts of a foot, and inches with their decimal parts; lengths of spans, and

. thicknesses of concrete, respectively, as `Just described for scale C above, is novel and a feature'possessed b no other sliderule of which I have know edge, and the advantage of which may be readily appreciated. p

On the scale D of the areas of metal the log. of A is laid out. 'n is computed and the log. laid out in scale B. On scale E the log. of moments is `laid out.

It is evident from equation I that the relative positions of 'w and Z being fixed, the

relative positions of m and Z are also fixedl and,'if of m and Z the latter d is fixed by using the same division points as are used for Z, then the position of m is absolutely fixed with respect to fw. It is also' evident from equation II that if the-relative positions of d and A are fixed, then the position of n is absolutely fixed. Now since m and 'irmay be computed for any formula in common use, it is therefore evident that my rule is not limited to any one formula. It is further evident that ifthe relative positions of the scales F, C and D are maintained, that innumerable values of m and n may be located in scales A and B, these values of m and n being determined by any selected values for the terms involved in the primary equations; it is thus seen that any equations in usein solving problems in l,connection with reinforced concrete, may be solved by the proper location of the pointsmandninthe scales A and B, and my rule is therefore applicable to any formula.

In the rule shown in Figs. 1 and 2, the following values are used in determining m and n:

0:1500 and variable.

R:12 and variable.

1:32 and variable.

6:12 inches.

Z:span in feet.

w:load in pounds per square foot.

M:moment in foot pounds per foot wide.

A: area of metal in square inches per foot wide of beam.

Since fm, and 'n are functions of lr', the ratio of the unit stresses, and 1' a function of p, the percentage of reinforcement,` the points computed for m and n may be designated by either the ratios r or the percent- Vages p, for which the are computed, or the ratios may be sub ivided into the unit stresses that produce them, and these unit stresses used to designate these subdivisions, in the lines marked A ratios and B ratios respectively. If now the load per square foot be brought over the span in feet, the depth of concrete may be read under any ratio of unit stresses or percentage of reinforcement desired and shown on the A ratio, and the area of metal read over the same ratio or percentage shown on. the B ratio. If the loading is not uniform, then the bending moment may be computed. In this case the general formula M:g w Z2 represents this formula, 'wf being the equivalent uniform load. Putting this into logarithm form we have:

1/2 10g. M i/a iogg'gi/a-iog. w+10g. z, g being a constant this reduces to 1/2 log. M-,Kzl/SZ log. lw-t-log. Z.

On scale F 1/2 log. lw has been laid out and log. Z on scale C, hence by laying E Fig. 1.

out 1/2log. -M-K, on some other scale, say E, then the relative positions of these scales having been fixed a reading point P is absolutely fixed. Hence, should the load be not uniform the bending moment of the external forces is computed and the point designated as P on scale D in Fig. 1 is brought over the bending moment on scale The thickness of concrete and the area of metal for any selected ratio may then. be read as in the case of a uniform load as already--idescribed In the use of this scale ofbending moments, it is to be noted that the uniform load on a free span equivalent to the load or loads producing any bending moment may be read over any specified span, and this irrespective of the material or materials composing the structure.

A secondar scale C2 of unit stresses for concrete is laid out for the purpose of determining any function for a unit stress other than 'that for which a rule may be particularly constructed. Its use for this purpose is more fully shown hereinafter.

The rule shown in Figs. 1\, 2 and 3 has been computed for beams simply supported at the ends, but the same method of laying out can be designed for continuous beams. This is not deemed necessary, as if, it is desired to consider continuity the rule as shown may also be used without change of form for the reason that a continuous beam carrying the same load per square foot as a beam simplysupported will have the same maximumstresses, as the latter if the span of the latter is considered as 9/10 of the continuous, for example; by reducing any given span of a continuous beam by ,1/10, the rule as shown may be used for the new span thus found. Again the rule may be used as shown b those whose practice it is" to arbitrarily ta e the clear span as the span to be computed, on the assumption that this method fully compensates for the difference between simply supported and continuous beams. Or the. bending moment fory any 4manner of loading and for any end conditions maybe com uted, and the rule used as shown. Ori the load per square foot on the continuous beam be reduced 20%, and the'new load so determined used as before.

It is noted that the rule as shown in Figs. 1 and 2 and the formulae used are for determining the thickness of concrete from outer ber to center of embeddedv metal. This in ordinary practice is the depth first obtalned, a sufficient amount of concrete being added to properly embed and protect the size of metal selected for use. If, however, it 1s deemed v desirable 1 to determine the total -*depth volf concrete, it is ordinarily assumed that one depthis afunction of the other. For example the distance from outer ber to center of metal is arbitrarily 'taken as 9/10 the total depth and -in this way the rule may be designed,r by a slight modification to give the total depth instead of the depth as shown. It is to beunderstood that in pointing out in this way the certain possible'calculations which can be made with this rule, no attempt is made to 've all the possible calculations, this ruleA eing, like most slide-rules, capable of determining any one or more of several factors when the remaining factors are known.

The following examples will illustrate some of the uses of my rule. Figs. 1, 2 and 3 in accompanying drawings show the rule as used in the solutions.

Example I: It is desired to design a slab of reinforced concrete, that will support a load of 225 pounds per square foot, which load also includes the Weight of slab, over a span of 7-6 with unit stresses of 500 pounds per square inch in concrete and 16000 pounds per square inch in the embedded metal, thus giving a ratio of Set 225 on scale of loads over 7.5 on scalel of types and sizes of metal from which may be selected any one desired, with the corresponding distances from center to center at which the onei selected must be placed to give the required area. For example:

7/16"round bars should be placed 7" on centers.

square 65/8 on centers. 5/16 4 1-/2 0n centers.

6" No. 4 expanded metal being nearest the ratio,32 would be selected as the size to be employed if that type is used. If the small size Kahn bar is used, it should-be spaced 16 on centers as shown.

Example II: Assuming that the load is notY uniform, but that the computed bending moment is 1570 foot pounds, and it is'required to'idesign a reinforced concrete slab with the same unit stresses as in Examplel I. Place point marked P on scale of areas over 1570, the computed bending moment, and proceed as in xample I.

Example III: It 1s required to determine how much a slab having a depth of 4 3/4 from top of outer fiber to center of embed` ded metals having an area of .325, per ft.

in width of concrete, will support over vari- `reverse side of rule will be found a variety ous spans. The slide is moved until the A I .cludes the weight of slab. For instance over a span of 7 6 (7.5) will befound 225 lbs. per square foot as the load; over a span of 8 ft. will be found 200 lbs., nearly and so on.

Example III may also be solved as follows:

to determine the percentage of metal which is found to be .57 nearly, which is the percentage fo'iind under 27 (nearly) on the scale of B ratios. If it is desired to know. the unit ,stresses the ratios 27 in the foregoing example corresponding to .57% reinforcement gives unit stresses of 500 lbs. per square inch on the concrete and 27 X500: 13500 lbs. per square inch on the steel. If 16000 lbs. per square inch were desired with the same ratio 27, or percentage ofl reinforcement .57, the load determined above (Example III) would then be multiplied by and the stress on the concrete would be p X 500 592 pounds per square inch.

Again, should the stress on the concrete be limited to an amount other than 500 that for which the rule illustrated is constructed, the other functions remaining the same as in foregoing Example III, proceed as in Example III, then note the reading on the scale E of moments under the point P on the scale D and bring 400 on scale C2 4to this reading on scale E. The required load will then be formed over any span as :-180 lbs. per square foot for a span of 7 6.

The selection of classes of reinforcement and their respective spacings when the B ratio differs from S32-for which the rule illustrated was constructedis determined by placing the area as found by -the ratio used over 32 B ratio and the types and distances from centers will then be found as previously described in- Example I.

proceed as before with the area so det-ermined.

While I have shown and described certain forms and arrangement of rules and scales, it isl to be understood that variations and modifications can be made without departing from the value and principle of the invention.

lVhat I claim as new and desire to secure by Letters Patent is as follows:

l. A logarithmic scale for designing and investigating the design of reinforced concrete beams or the like consisting of two relatively movable scale members, one of which is provided with two independent similarly laid off logarithmic scales of ratios of unit stresses and a logarithmic scale of loads per square feet in pounds and the other of which scale members lis provided with a logarithmic scale common to one of the ratio scales and the load scale of the first scale member and indicating when read on the ratio scale the depth in inches of concrete required for a beam of a certain ratio to support a given load and when read on the load scale the length in `feet of the beam required to support such load, and with a scale of area of steel in square inches per foot wide of concrete arranged to cooperate with the other ratio scale on the lfirst mentioned scale member, all of said scales being arranged to permit readings from a single adjustment of the scale members.

.with a logarithmic scale of loads per square foot in. ounds, and with a logarithmic scale of ben ing moments in foot pounds per foot wide of concrete, the other of said scale members being provided with a combined logarithmic scale of span in feet and thickness in concrete in inches common to one of the ratio scales and the load scale on the first mentioned scale member, and with a scale of area of steel in square inches er footrwide of concrete coi'nmon to the ot er ratio scale and scale of bending moments of said first mentioned scale members, all of said scales being arranged to permit readings from a single position on the scale.

3. A logarithmic scale for d esignln and investigating the design of reinforce concrete consisting of two relatively movable scale members, one of whih is provided [with two""independent 'similarly laid olf members being provided with a combined logarithmic scale of span in feet and thickness in concrete in inches common to one of the ratio scales and the load scale on the first mentioned scale member, and with a scale of area of steel in square inches er foot wide of concrete common t'o the ot 1er ratio scale and scale of bending moments of said rst mentioned scale members, all of said scales being arranged to permit readings from a single osition on the scale, one of said scale mem ers being further provided with a scale of sizes-of reinforcing materials and 'the other of said scales with a distance scale coperating with said scale -of sizes, said dist-ance scale indicating after the scale1 members have been adjusted to determine the reinforcement re uired in any beam, the distance apart any o the material in the material scale must be spaced to obtain such amount of reinforcement.

4.' A logarithmic scale comprisin two relatively movable scale members, sai scale members being provided with a plurality of coperating lo arithmic scales enabling the determination y lthe adjustment of one of the scale membersy of the amount of reinforcement required in a beam of known size to support a known load, and said scale members being also provided one with a scale of different; sizes of reinforcing materials and the other with a coperatlng distance scale, such scales being so arranged that when the scale members'of therule have been adjusted to ascertain the amount of reinforcement required, the distance scale may b'e read on the material scale to indicate the dispositionv of any of the -materials therein to obtain such certain reinforcement.

5.k A slide rule comprising separated face plates having openings therein, said face plates being provided with scales adjacent said openings, and a slide positioned between the face plates and provided with scales which are visible through the openings in the face plates and which are adapted to coperate with the scales adjacent thereto.

6. A slide rule comprising a face plate having an opening therein one edge of which comprises a plurality of oset portions, said face plate being provided with scales adjacent the edges of the opening, and a slide underlying the face plate provided with scales which are visible through the opening in the face plate and which are adapted to coperate with the scales adjacent the edges of said opening.

7 A slide rule com rising parallel side bars with proximate e ges squared, a plate connecting said side bars and overlying the faces thereof, said plate having an opening therein, and a slide havingv squared edges positioned between said side bars.

8. A slide rule comprising parallel side bars having their proxlmate edges squared, face plates connecting said side bars and overlying the u per and lower faces thereof, and a slidehavlng squared edges positloned between said side bars. f

9. A slide rule comprising parallel bars having their proximate edges squared, face plates overlying the upper and lower surfaces of said side bars and provided with scales, cross bars connecting the ends of the parallel bars and clamping the face plates thereto, and a slide having squared edges positioned between said side bars,

' said slide being provided with a-scale adapted to coperate with the scale of the face plates.

In testimony whereof aix my signature in presence of two witnesses.

JOSEPH MOC. MICHAELSON.

Witnesses?) I RICHARD EznoRr, EGBERT P. LINCOLN. 

